In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to that of Jean Louis Koszul?
3
-
2$\begingroup$ Kobayashi-hyperbolicity is defined for complex manifolds. What is Koszul's definition for affine-flat(?) manifolds? $\endgroup$– Moishe KohanCommented Sep 25, 2023 at 18:22
-
$\begingroup$ A closed locally flat manifold (affine) is hyperbolic in the sense of Koszul if its universal covering is affinely diffeomorphic to a sharply convex cone in ℝⁿ. Sharp convex means a convex domain in ℝⁿ containing no lines. $\endgroup$– SamirCommented Sep 25, 2023 at 18:47
-
1$\begingroup$ Well, then the answer to your question is obviously negative. You just have to understand the definition and some basic properties which you find in any source discussing Kobayashi-hyperbolicity. $\endgroup$– Moishe KohanCommented Sep 25, 2023 at 18:52
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
This might help clarify why the term hyperbolicity is ubiquitous. Let $K$ be a compact set in $\mathbf{R}^n$, we say that $K$ is convex if for any two points $x,y \in K$, the line $\{ (1-t) x + ty : 0 \leq t \leq 1 \}$ is contained in $K$. This describes a form a "geometric connectedness" of the set $K$. Notice that convexity as defined here, is equivalent to the fact that any two points $x,y \in K$ are contained in the image of an affine map $A : [0,1] \to K$. Hence, ordinary geometric convexity describes a connectedness with respect to affine maps from the compact unit interval. In this way, convexity can be generalized in many ways, from transitive group actions, to rationally connected manifolds, the latter defined by replacing $[0,1]$ with $\mathbf{CP}^1$, $K$ with a (compact) complex manifold $X$, and affine maps with holomorphic maps $\mathbf{CP}^1 \to X$.
Hyperbolicity is given by violating this convexity criterion completely. For (compact) Kobayashi hyperbolic manifolds, this translates to every holomorphic map $\mathbf{C} = \mathbf{CP}^1 \backslash \{ \infty \} \to X$ being constant.
Of course, as Moishe Kohan points out, Kobayashi hyperbolicity is very different to this notion of hyperbolicity due to Koszul.
